Switched linear singular systems in discrete time: Solution theory and observability notions

We study the solution theory of linear switched singular systems. In a recent paper by Anh et al. (2019), it was highlighted that the common assumption that each mode of the switched system is index-1 is not sufficient to guarantee existence and uniqueness of solutions of the corresponding switched system and the notion of jointly index-1 was introduced. However, until now it was not clear what conditions are actually required to guarantee existence and uniqueness of solutions if the switching signal is not considered arbitrary. In particular, we study the two relevant situations where the mode sequence is fixed (and the switching times are arbitrary) and where the whole switching signal is fixed. In both cases, we provide conditions in terms of the original system matrices which ensure existence and uniqueness of solutions. We also extend the idea of the one-step map introduced by Anh et al. (2019) to these two cases. It turns out that in the case of a fixed switching signal, the index-1 condition for the individual modes is also not necessary (in addition to not being sufficient). Furthermore, we utilize the established solution theory to provide characterizations of observability and determinability of switched singular systems.